where M is a Matrix (for example a 3x3), x is a vector (3 components)
and b is a real number (could also be complex number).
You see that the Matrix doesn't change the direction of x, only it's length (right hand side of the equation).
x is called eigenvector and b eigenvalue of M.

Now in Quantum mechanics you have operators (instead of matrices)
and so called state vectors,

for example:

H |Psi> = E |Psi>

( M x = b x )

H is the Hamilton-Operator, |Psi> is your eigenvector and E the eigenvalue.

Whats the meaning of the equation above?
It just says that you got a system represented by the vector |Psi>
(for example electron in the Hydrogen atom).
And then you want to measure the energy. This is done by
'throwing' the operator H on your vector |Psi>. What comes out
is your eigenvalue E which is the energy.

Now whats the Schrödinger equation?
Suppose you want to examine the energy of the electron in the hydrogen atom. So you just apply H on |Psi> and get the energy E on the right hand side of the eigenequation.
The PROBLEM is, you dont know how your |Psi> looks like.

So here's where the SCHRÖDINGER equation comes into the play.
The Schrödinger equation is a differential equation,
which you have to solve in order to get your |Psi>. (solving the differential equation means you get a solution |Psi>)

You put your potential (square well potential for particle in a box, or Coloumb potential for hydrogen atom) into the Schrödinger equation and solve it. You get your |Psi> from it.